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There is hidden linear time to perform merge part.
Linear is less than loglinear, so it was not visible. $log1$ is equal 0, but it means that you will not e.g. heapify, just merge in O(n).
Also all other complexities hide constant factors, so $O(logk + c)$ drops constant part, but when it becomes 0, you should reveal hidden part and say it is $O(1)$.